The tone in a discussion is the general feeling, mood, or attitude of the discussion. Just as it is an analogy to a musical tone, tone can be raised and lowered.
Raising the tone of a discussion or argument is to increase or heighten the intensity or intellectual rigour and thereby raising the standard of discussion.
Setting the tone establishes the feeling or mood of an event or happening.
His opening remarks set the tone for the rest of the discussion.
Her superfluous remarks lowered the tone of the discussion.
The experience and real-world examples raised the tone of the discussion.
I believe the tone referred to in the NYT article was the debate on the intellectual rigour and character of Arendt in general and possibly Arendt's defence of Heidegger's actions as a Nazi sympathizer in particular, just after Ettinger's publication..
After Ettinger's publication, Arendt's critics
seized on it as evidence that she was intellectually untrustworthy
possibly because of the close relationship she had with Heidegger, her critics may have felt she was compromised by love. Arendt's defenders
were not slow to respond
in defence of Arendt (whom they thought of in a saintly way), meaning they defended her as well as they could, but
did little to raise the tone
left implicit is the argument of Arendt's credibility. I believe Arendt's critics resorted to something similar to tabloid sensationalism in attacking her character and Arendt's defenders were not able to respond since only a handful of people had actually seen the letters.
Once the letters were generally published
it more than set the record straight
showing the intellectual strength of Arendt, Heidegger, and Jaspers relationship, regardless of their personal intertwining.
[ BTW: Interesting article, thanks ]